Samiksha Jaiswal (Editor)

Universal coefficient theorem

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In algebraic topology, universal coefficient theorems establish relationships between homology and cohomology theories. For instance, the integral homology theory of a topological space X, and its homology with coefficients in any abelian group A are related as follows: the integral homology groups

Contents

Hi(X; Z)

completely determine the groups

Hi(X; A)

Here Hi might be the simplicial homology or more general singular homology theory: the result itself is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients A may be used, at the cost of using a Tor functor.

For example it is common to take A to be Z/2Z, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers bi of X and the Betti numbers bi,F with coefficients in a field F. These can differ, but only when the characteristic of F is a prime number p for which there is some p-torsion in the homology.

Statement of the homology case

Consider the tensor product of modules Hi(X; Z) ⊗ A. The theorem states there is a short exact sequence

0 → H i ( X ; Z ) ⊗ A → μ H i ( X ; A ) → Tor ⁡ ( H i − 1 ( X ; Z ) , A ) → 0.

Furthermore, this sequence splits, though not naturally. Here μ is a map induced by the bilinear map Hi(X; Z) × A → Hi(X; A).

If the coefficient ring A is Z/pZ, this is a special case of the Bockstein spectral sequence.

Universal coefficient theorem for cohomology

Let G be a module over a principal ideal domain R (e.g., Z or a field.)

There is also a universal coefficient theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence

0 → Ext R 1 ⁡ ( H i − 1 ⁡ ( X ; R ) , G ) → H i ( X ; G ) → h Hom R ⁡ ( H i ( X ; R ) , G ) → 0.

As in the homology case, the sequence splits, though not naturally.

In fact, suppose

H i ( X ; G ) = ker ⁡ ∂ i ⊗ G / im ⁡ ∂ i + 1 ⊗ G

and define:

H ∗ ( X ; G ) = ker ⁡ ( Hom ⁡ ( ∂ , G ) ) / im ⁡ ( Hom ⁡ ( ∂ , G ) ) .

Then h above is the canonical map:

h ( [ f ] ) ( [ x ] ) = f ( x ) .

An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map h takes a homotopy class of maps from X to K(G, i) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.

Example: mod 2 cohomology of the real projective space

Let X = Pn(R), the real projective space. We compute the singular cohomology of X with coefficients in R = Z/2Z.

Knowing that the integer homology is given by:

H i ( X ; Z ) = { Z i = 0  or  i = n  odd, Z / 2 Z 0 < i < n ,   i   odd, 0 else.

We have Ext(R, R) = R, Ext(Z, R) = 0, so that the above exact sequences yield

∀ i = 0 , ⋯ , n :   H i ( X ; R ) = R .

In fact the total cohomology ring structure is

H ∗ ( X ; R ) = R [ w ] / ⟨ w n + 1 ⟩ .

Corollaries

A special case of the theorem is computing integral cohomology. For a finite CW complex X, Hi(X; Z) is finitely generated, and so we have the following decomposition.

H i ( X ; Z ) ≅ Z β i ( X ) ⊕ T i ,

where βi(X) are the Betti numbers of X and T i is the torsion part of H i . One may check that

Hom ⁡ ( H i ( X ) , Z ) ≅ Hom ⁡ ( Z β i ( X ) , Z ) ⊕ Hom ⁡ ( T i , Z ) ≅ Z β i ( X ) ,

and

Ext ⁡ ( H i ( X ) , Z ) ≅ Ext ⁡ ( Z β i ( X ) , Z ) ⊕ Ext ⁡ ( T i , Z ) ≅ T i .

This gives the following statement for integral cohomology:

H i ( X ; Z ) ≅ Z β i ( X ) ⊕ T i − 1 .

For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that βi(X) = βn−i(X).

References

Universal coefficient theorem Wikipedia
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